Mathematical Explanation in the NYRB

In his recent review of Dawkins' Oxford Book of Modern Science Writing Jeremy Bernstein characterizes one entry as follows:

W.D. Hamilton’s mathematical explanation of the tendency of animals to cluster when attacked by predators.

The article in question is "Geometry for the Selfish Herd", Journal of Theoretical Biology 31 (1971): 295-311. (Online here.) Given the ongoing worries about the existence and nature of mathematical explanations in science, it is worth asking what led Bernstein to characterize this explanation as mathematical?

The article summarizes two models of predation which are used to support the conclusion that the avoidance of predators "is an important factor in the gregarious tendencies of a very wide variety of animals" (p. 298). The first model considers a circular pond where frogs, the prey, are randomly scattered on the edge. The predator, a single snake, comes to the surface of the pond and strikes whichever frog is nearest. Hamilton introduces a notion of the domain of danger of a frog which is the part of the pond edge which would lead to the frog being attacked. Hamilton points out that the frogs can reduce their domains of danger by jumping together. In this diagram the black frog jumps between two other frogs:



So, "selfish avoidance of a predator can lead to aggregation."

In the slightly more realistic two-dimensional case Hamilton generalizes his domains of danger to polygons whose sides result from bisecting the lines which connect the prey:



Hamilton notes that it is not known what the general best strategy is here for a prey organism to minimize its domain of danger, but gives rough estimates to justify the conclusion that moving towards ones nearest neighbor is appropriate. This is motivated in part by the claim that "Since the average number of sides is six and triangles are rare (...), it must be a generally useful rule for a cow to approach its nearest neighbor."

So, we can explain the observed aggregation behavior using the ordinary notion of fitness and an appeal to natural selection. What is the mathematics doing here and why might we have some sort of specifically mathematical explanation? My suggestion is that the mathematical claim that strategy X minimizes (or reliably lowers) the domain of danger is a crucial part of the account. Believing this claim and seeing its relevance to the aggregation behavior is essential to having this explanation. Furthermore, this seems like a very good explanation. What implications this has for our mathematical beliefs remains, of course, a subject for debate.

Southern Journal of Philosophy Relaunched

The Southern Journal of Philosophy has relaunched with a new publishing agreement with Wiley, a new webpage and a new editorial board (including me). As the webpage indicates

The Southern Journal of Philosophy has long provided a forum for the expression of philosophical ideas and welcomes articles written from all philosophical perspectives, including both the analytic and continental traditions, as well as the history of philosophy. This commitment to philosophical pluralism is reflected in the long list of notable figures whose work has appeared in the journal, including Hans-Georg Gadamer, Hubert Dreyfus, George Santayana, Wilfrid Sellars, and Richard Sorabji.

The jewel of each volume is the Spindel Supplement, which features the invited papers and commentaries presented at the annual Spindel Conference. Held each autumn at the University of Memphis and endowed by a generous gift from the Spindel family, each Spindel Conference centers on a philosophical topic of broad interest and provides a venue for discussion by the world's leading figures on that topic.

I hope the philosophers will take advantage of this special venue for pursuing new and exciting directions for research in philosophy.

New Entries in Internet Encyclopedia of Philosophy on the Philosophy of Mathematics

Under the editorial guidance of Roy Cook a number of new entries in philosophy of mathematics have appeared on the Internet Encyclopedia of Philosophy. As I understand it, the aim of this site is to present relatively short summaries which are accessible to a wider audience, esp. undergraduate students, than some other options.

Check out these recent entries:

Bolzano's Philosophy of Mathematical Knowledge (by Sandra Lapointe)

The Applicability of Mathematics (by me -- more shameless self-promotion!)

Mathematical Platonism (by Julian Cole)

Predicative and Impredicative Definitions (by Oystein Linnebo)


A list of the all of the philosophy of mathematics entries can be monitored here.

Wash Post Reminds Us That There is No Perfect Climate Model

Here. There are some useful quotations from scientists, including:

If the models are as flawed as critics say, Schmidt said, "You have to ask yourself, 'How come they work?'"

What is missing from the article, though, is any discussion of the more or less risky claims which we might derive from examining a model or computer simulation. It seems that even though the models are highly detailed, most climate scientists are comfortable drawing only highly abstract conclusions. For example, they do not take seriously the temperature predictions for Indiana, but do take seriously the predications for the global mean temperature.

Inference to the Best Explanation for Mathematical Claims

Following up my last post, I want to outline an argument for why a reasonable restriction on IBE will block the use of IBE to provide much justification for mathematical claims. The main "mathematical explanations" which are discussed by advocates of this sort of justification, like Baker and Colyvan, involve explanations of patterns observed in the physical world. These examples include the life-cycle of cicada and the shape of the cells of a honeycomb. One of the problems with these explanations is that they bring in complications associated with explanations via natural selection. Another problem is that they may involve mathematical terms in the description of what is to be explained.

To avoid these problems, I will focus on the bridges of Konigsberg case (see here for some background). The explanation could be reconstructed as

(1) The bridges of Konigsberg form a graph of type O.
(2) There is no Euler path through a graph of type O.
(3) Therefore, there is no Euler path through the bridges of Konigsberg.

An Euler path is a circuit through the graph that crosses each edge exactly once. For someone who worries that even this begs the question by using a mathematical term we can offer to extend the explanation to include "(4) Therefore, it is impossible to cross each of the bridges exactly once."

I claim that Sensitivity blocks the use of IBE to support (2). This is because an agent who was genuinely in doubt about the truth of (2) would also have as a relevant epistemic possibility that (2') There is no Euler path through a graph of type O with fewer than 100 vertices. This means that there is an alternative explanation of (3) which employs weaker mathematical assumptions:

(1’) The bridges of Konigsberg form a graph of type O with fewer than 100 vertices.
(2’) There is no Euler path through a graph of type O with fewer than 100 vertices.
(3) Therefore, there is no Euler path through the bridges of Konigsberg.

My conclusion, then, is that this puts the burden on the advocates of using IBE to justify mathematical claims to argue that Sensitivity is incorrect or that some other features of these cases have been overlooked.

Inference to the Best Explanation and Sensitivity

Philosophers of science have focused on inference to the best explanation (IBE) as the sort of inference that stands the best chance of ultimately justifying our belief in unobservable entities like atoms and electrons. More recently philosophers of mathematics like Colyvan and Baker have tried to given an explanatory indispensability argument in support some of our mathematical beliefs. The challenge for everyone, though, is to articulate a reasonable form of IBE that accords with scientific practice, but which does not overgenerate beliefs in things which we reject. Despite its clarity, Lipton's discussion of IBE seems to me overly restrictive because he focuses only on causal explanations. Are there plausible principles for the use of IBE which allow non-causal explanations?

Here is one, but it results in problems for any use of IBE to justify our mathematical beliefs. I call it "Sensitivity", although perhaps this not the best label:

Sensitivity: A claim which appears in an explanation can receive support via IBE only when the explanatory contribution tells against some relevant epistemic possibilities.

Here I am imagining an agent who is in doubt about the truth of some competing options A1, A2, A3. Suppose that A1 appears in our best explanation. Sensitivity tells us that this contribution of A1 to the explanation can only license belief in A1 when the way in which A1 contributes makes either A2 or A3 less likely.

This seems to me to be a very weak and plausible restriction on IBE. It is met by the standard atoms and electrons cases, and also by Woodward's non-causal explanation of the stability of planetary orbits. In my next post, I want to outline a case for the claim that sensitivity blocks the use of IBE to support mathematical claims.

Book Project Update

For those few readers tracking my ongoing book project on Mathematics and Scientific Representation some recent good news is that I have signed a contract with Oxford University Press. The delivery date in the contract is Nov. 2010, so over the next six months I will posting some of the key ideas, and eventually the near-final versions of the chapters, for comments and discussion.

The manuscript is projected to be 140 000 words, with twelve chapters:

1. Introduction

Part I: Epistemic Contributions

2. Content and Confirmation
3. Causes
4. Varying Interpretation
5. Scale Matters
6. Constitutive Frameworks
7. Failures

Part II: Other Contributions

8. Discovery
9. Indispensability and Explanation
10. Fictionalism
11. Facades

12. Conclusion: Pure Mathematics